Algorithm for detecting repeating decimals?

Question

Is there an algorithm for figuring out the following things?

1. If the result of a division is a repeating decimal (in binary).
2. If it repeats, at what di

Answer 1

You can do a long division, noting the remainders. The structure of the remainders will give you the structure of any rational decimal:

1. the last remainder is zero: it is a decimal without any repeating part
2. the first and the last remainder are equal: the decimal is repeating right after the dot
3. the distance between the first and the first remainder equal to the last are the non-repeating digits, the remainder is the repeating part

In general the distances will give you the amount of digits for each part.

You can see this algorithm coded in C++ in the method decompose() here.

Try 228142/62265, it has a period of 1776 digits!